In the comments on Ralph’s post, Julien raised the issue of Glimpse cases as a problem for adherence conditions on knowledge, in response to my question about whether a probabilistic adherence condition would be advantageous to use. Glimpse cases, such as glimpsing a painted bunting in my live oak tree as I happen to look up from writing this post, present challenges to adherence conditions, since one is highly likely not to have noticed the bird at all.
I think Roush’s probabilistic truth-tracking theory is much superior to the original Nozick theory, so wanted to post on whether her account can handle Glimpse cases. The feature of her theory that must do the work here is the condition on what gets held fixed when assessing the relevance condition and the adherence condition, which are, respectively:
P(~B(p)|~p) is very high.
P(B(p)|p) is very high and P(B(~p)|p) is very low.
I’ll put the details of what gets held fixed and how this helps with Glimpse cases below the fold, but the interesting point here is that the fixing condition allows the conclusion that Roush’s adherence condition is satisfied in Glimpse cases.
So, first the technical details of what gets held fixed. What gets held fixed is the actual (objective) probability of a claim, given (and only given) that it meets either of two inequality conditions. I’ll given these formally, and then give Roush’s informal gloss on them. The formal inequalities are these:
For all q such that q is not a conjunction or a universal generalization nor B(p), ~B(p), B(p), ~B(~p); hold Pr(q) fixed at its actual value iff:
* $$|Pr(q|\sim\!p) – Pr(q|p)| \leq |Pr(\sim\!p|q) – Pr(p|q)|$$ and $$|Pr(\sim\!q|\sim\!p) – Pr(\sim\!q|p)| \leq |Pr(\sim\!p|q) – Pr(p|q)|$$
** $$|Pr(q|p) – Pr(q|\sim\!p)| < |Pr(p|q) - Pr(p|\sim\!q)|$$ and $$|Pr(\sim\!q|p) - Pr(\sim\!q|\sim\!p)| < |Pr(p|q) - Pr(p|\sim\!q)|$$. Roush glosses these conditions as follows: "Criterion * says that p's being false rather than true makes less or the same difference to q's being true, and to q's being false, than the difference that q's being true or false makes to p's being false." For criterion **, she says, "[H]ere we want to allow variation in those matters that the truth value of p makes no difference to, and allow variation in those things that the truth of p makes more of a difference to than they make to p." The condition useful in responding to Glimpse cases is criterion *, and notice the presence of the painted bunting in my tree makes no probabilistic difference to whether I glance in the way I did or whether I didn't glance at all. Notice as well that whether I glance in the way I did, or whether I didn't glance, makes no difference to whether there is a painted bunting in my live oak tree. So, criterion * is satisfied in this case for the claim about the particularities of my glimpse (q), because p's being false rather than true makes the same difference to the truth value of q as the truth value of q makes to p. So the actual probability of q gets held fixed when assessing the adherence condition, and the actual probability of the particularities of my glimpse is, by hypothesis, 1. So, the probability involved in the adherence condition is the probability of my believing that there is a painted bunting in my live oak tree, given the presence of the bird and the particularities of my glimpse. This probability strikes me as plausibly held to be very high. (I will here assume minimal logical coherence for my own belief states, so the probability of believing not-p given p will be correspondingly low--this assumption is an important one in Roush's theory, and generates independent concerns about the theory, but since they are independent of the worries caused by Glimpse cases for adherence conditions on knowledge, I'll ignore them here. If you wish further info on that issue, see Horacio's review of Sherri's book at NDPR). So, bottom line: it looks like Roush's fixing condition gives a decent answer to how the adherence condition is satisfied in Glimpse cases.